Junior problems - Phần 4

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Junior problems - Phần 4 Junior problems J181. Let a, b, c, d be positive real numbers. Prove that 3 3 3 3 a2 + d2 b2 + c2 a+b c+d ≤ + + 2 2 a+d b+c Proposed by Pedro H. O. Pantoja, Natal-RN, Brazil J182. Circles C1 (O1 , r) and C2 (O2 , R) are externally tangent. Tangent lines from O1 to C2 intersect C2 at A and B, while tangent lines from O2 to C1 intersect C1 at C and D. Let O1 A ∩ O2 C = {E } and O1 B ∩ O2 D = {F }. Prove that EF ∩ O1 O2 = AD ∩ BC. Proposed by Roberto Bosch Cabrera, Florida, USA J183. Let x, y, z be real numbers. Prove that 2 (x2 + y 2 + z 2 )2 + xyz (x + y + z ) ≥ (xy + yz + zx)2 + (x2 y 2 + y 2 z 2 + z 2 x2 ). 3 Proposed by Neculai Stanciu, George Emil Palade, Buzau, Romania J184. Find all quadruples (x, y, z, w) of integers satisfying the system of equations x + y + z + w = xy + yz + zx + w2 − w = xyz − w3 = −1. Proposed by Titu Andreescu, University of Texas at Dallas, USA 2xy J185. Let H (x, y ) = x+y be the harmonic mean of the positive real numbers x and y . For n ≥ 2, find the greatest constant C such that for any positive real numbers a1 , . . . , an , b1 , . . . , bn the following inequality holds C 1 1 ≤ + ··· + . H (a1 + · · · + an , b1 + · · · + bn ) H (a1 , b1 ) H (an , bn ) Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania J186. Let ABC be a right triangle with AC = 3 and BC = 4 and let the median AA1 and the angle bisector BB1 intersect at O. A line through O intersects hypotenuse AB at M and AC at N . Prove that MB NC 4 · ≤. MA NA 9 Proposed by Valcho Milchev, Kardzhali, Bulgaria 1 Mathematical Reflections 1 (2011) Senior problems S181. Let a and b be positive real numbers such that 1 1 |a − 2b| ≤ √ and |2a − b| ≤ √ . a b Prove that a + b ≤ 2. Proposed by Titu Andreescu, University of Texas at Dallas, USA S182. Let a, b, c be real numbers such that a > b > c. Prove that for each real number x the following inequality holds 1 (x − a)4 (b − c) ≥ (a − b)(b − c)(a − c)[(a − b)2 + (b − c)2 + (c − a)2 ]. 6 cyc Proposed by Dorin Andrica, Babes-Bolyai University, Cluj-Napoca, Romania a2 −1 n S183. Let a0 ∈ (0, 1) and an = an−1 − n ≥ 1. Prove that for all n, 2, √ n 1 1 n+1+ n ≤ − < . 2 an a0 2 Proposed by Arkady Alt, San Jose, California, USA 1 1 1 + · · · + n , n ≥ 2. Prove that S184. Let Hn = + 2 3 √ n eHn > n! ≥ 2Hn . Proposed by Tigran Hakobyan, Vanadzor, Armenia S185. Let A1 , A2 , A3 be non-collinear points on parabola x2 = 4py, p > 0, and let B1 = l2 ∩ l3 , B2 = l3 ∩ l1 , B3 = l1 ∩ l2 where l1 , l2 , l3 are tangents to the parabola [ A1 A2 A3 ] at points A1 , A2 , A3 , respectively. Prove that ...